Mathematics

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1500 questions
67
votes
3 answers

Closed form solution for $\sum_{n=1}^\infty\frac{1}{1+\frac{n^2}{1+\frac{1}{\stackrel{\ddots}{1+\frac{1}{1+n^2}}}}}$.

Let $$ \text{S}_k = \sum_{n=1}^\infty\cfrac{1}{1+\cfrac{n^2}{1+\cfrac{1}{\ddots1+\cfrac{1}{1+n^2}}}},\quad\text{$k$ rows in the continued fraction} $$ So for example, the terms of the sum $\text{S}_6$ are $$…
Simon Parker
  • 4,431
66
votes
5 answers

Picking random points in the volume of sphere with uniform probability

I have a sphere of radius $R_{s}$, and I would like to pick random points in its volume with uniform probability. How can I do so while preventing any sort of clustering around poles or the center of the sphere? Since I'm unable to answer my own…
MHK
  • 663
66
votes
18 answers

What exactly is a number?

We've just been learning about complex numbers in class, and I don't really see why they're called numbers. Originally, a number used to be a means of counting (natural numbers). Then we extend these numbers to instances of owing other people money…
user164061
  • 849
66
votes
6 answers

Why do differential forms have a much richer structure than vector fields?

I apologize in advance because this question might be a bit philosophical, but I do think it is probably a genuine question with non-vacuous content. We know as a fact that differential forms have a much richer structure than vector fields, to name…
Jia Yiyang
  • 1,143
66
votes
22 answers

Nuking the Mosquito — ridiculously complicated ways to achieve very simple results

Here is a toned down example of what I'm looking for: Integration by solving for the unknown integral of $f(x)=x$: $$\int x \, dx=x^2-\int x \, dx$$ $$2\int x \, dx=x^2$$ $$\int x \, dx=\frac{x^2}{2}$$ Can anyone think of any more examples? P.S.…
Aidan F. Pierce
  • 1,431
66
votes
3 answers

Expectation of Minimum of $n$ i.i.d. uniform random variables.

$X_1, X_2, \ldots, X_n$ are $n$ i.i.d. uniform random variables. Let $Y = \min(X_1, X_2,\ldots, X_n)$. Then, what's the expectation of $Y$(i.e., $E(Y)$)? I have conducted some simulations by Matlab, and the results show that $E(Y)$ may equal to…
jet
  • 763
66
votes
4 answers

Mathematicians don't quit, they fade away

Edit: This question is now closed for being not related to math, but many people pointed out that becoming an actuary is one of the most viable career path for someone with skills in pure math. Noone I've ever talked to knows what mathematicians do…
Brian Rushton
  • 13,395
66
votes
3 answers

Why is it worth spending time on type theory?

Looking around there are three candidates for "foundations of mathematics": set theory category theory type theory There is a seminal paper relating these three topics: From Sets to Types to Categories to Sets by Steve Awodey But at this forum ( MSE…
Hans-Peter Stricker
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66
votes
7 answers

How can I show that $\sqrt{1+\sqrt{2+\sqrt{3+\sqrt\ldots}}}$ exists?

I would like to investigate the convergence of $$\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\sqrt\ldots}}}}$$ Or more precisely, let $$\begin{align} a_1 & = \sqrt 1\\ a_2 & = \sqrt{1+\sqrt2}\\ a_3 & = \sqrt{1+\sqrt{2+\sqrt 3}}\\ a_4 & =…
MJD
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66
votes
6 answers

Is $\{0\}$ a field?

Consider the set $F$ consisting of the single element $I$. Define addition and multiplication such that $I+I=I$ and $I \times I=I$ . This ring satisfies the field axioms: Closure under addition. If $x, y \in F$, then $x = y = I$, so $x + y = I + I =…
Dan
  • 19,542
66
votes
3 answers

Why are certain PDE called "elliptic", "hyperbolic", or "parabolic"?

Why are the Partial Differential Equations so named? i.e, elliptical, hyperbolic, and parabolic. I do know the condition at which a general second order partial differential equation becomes these, but I don't understand why they are so named? Does…
The_Lazy_Panda
  • 779
66
votes
5 answers

Is the box topology good for anything?

In point-set topology, one always learns about the box topology : the topology on an infinite product $X = \prod_{i \in I} X_i$ generated by sets of the form $U = \prod_{i \in I} U_i$, where $U_i \subset X_i$ is open. This seems naively like a…
Nate Eldredge
  • 102,564
66
votes
12 answers

Are there mathematical concepts that exist in dimension $4$, but not in dimension $3$?

Are there mathematical concepts that exist in the fourth dimension, but not in the third dimension? Of course, mathematical concepts include geometrical concepts, but I don't mean to say geometrical concept exclusively. I am not a mathematician and…
tefisjb
  • 1
66
votes
7 answers

Compute polynomial $p(x)$ if $x^5=1,\, x\neq 1$ [reducing mod $\textit{simpler}$ multiples]

The following question was asked on a high school test, where the students were given a few minutes per question, at most: Given that, $$P(x)=x^{104}+x^{93}+x^{82}+x^{71}+1$$ and, $$Q(x)=x^4+x^3+x^2+x+1$$ what is the remainder of $P(x)$ divided by…
joeblack
  • 1,023
66
votes
11 answers

7 fishermen caught exactly 100 fish and no two had caught the same number of fish. Then there are three who have together captured at least 50 fish.

$7$ fishermen caught exactly $100$ fish and no two had caught the same number of fish. Prove that there are three fishermen who have captured together at least $50$ fish. Try: Suppose $k$ th fisher caught $r_k$ fishes and that we have…
nonuser
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